VTMO 2026 - Grade 6 - Semifinal - SET 2 - Self-study

Parallelogram ABCD has area 40 cm². E and F are the midpoints of AD and DC, respectively. Find the area of quadrilateral AEFC.

A 4 × 4 board has a lamp in each small square. Touching a lamp changes its state and the states of all lamps in squares sharing a side with its square: on becomes off and off becomes on. Initially all lamps are on. What is the fewest touches needed to turn them all off?

A teacher writes all three-digit numbers whose digits are distinct, have product 24 and have sum 11. How many numbers can the teacher write at most?

Fill the 3 × 3 board with the numbers 1 to 9 so that every row sum and every column sum is prime. The numbers 1, 2 and 4 are fixed as shown. Find the product of the numbers in the four shaded squares.

Clarification for this practice version: the printed single-answer question admits more than one listed product. Select all listed products that are possible; keep all the original conditions.

A fraction of the form $1/n$, where n is an integer at least 2, is called an Egyptian fraction. Write $4/15=1/a+1/b$, with both summands Egyptian fractions. Find the smallest possible value of a + b.

Let P be the smallest composite number that is not divisible by any of the integers from 2 to 50. Find the sum of the digits of P.

Bình’s family has four people: father, mother, An and Bình. Their present ages satisfy these conditions: the average age of father and mother is 32; the average age of An and Bình is 8; the average age of father and Bình equals that of mother and An. How many years older is mother than Bình?

The figure is a 5 × 5 board. What is the smallest number of additional unit squares that must be shaded so that the resulting figure has exactly two axes of symmetry?

When 1001 is divided by an odd prime, the remainder is 1. What is the remainder when 2012 is divided by that prime?

Five football teams play a single round-robin tournament. A win gives 3 points, a loss gives 0, and a draw gives each team 1 point. After some matches, every team has positive points and all five point totals are different. What is the fewest matches that could have been played?

Find the units digit of $7^{2014}$.

The two diagonals of a 3 × 3 square grid pass through 5 small squares, as shown. Through how many small squares do the two diagonals of a 9 × 9 square grid pass?

Rectangle ABCD is divided into rectangles AEFD and BCFE as shown. The perimeter of BCFE is twice that of AEFD, and its area is four times that of AEFD. Given AD = 10 cm, find AB in centimetres.

A person takes 165 minutes to saw a 72 cm iron bar into equal pieces of length 6 cm. How long is needed to saw a 144 cm bar into 6 cm pieces?

A runner and a cyclist start together at point A and move clockwise around a circular lake. The cyclist travels at 7 times the runner’s speed. Each time they meet, their position is marked. How many distinct positions are marked?

The figures follow the pattern shown. How many black dots are there in the tenth figure?

What is the greatest possible number of Sundays in three consecutive calendar months?

Three drainage pipelines have equal length. The first consists of one 16 dm section and some 11 dm sections. The second consists of three 14 dm sections, nine 13 dm sections, and some 11 dm sections. The third consists of fifteen 20 dm sections and some 30 dm sections. Each pipeline is between 370 dm and 400 dm long. How many 11 dm sections are used altogether?

Châu builds the solid shown from small cubes, then paints its entire outer surface, including the bottom. Each face of a small cube needs 1 ml of paint. How many millilitres of paint are needed?

Tank A is full of water and tank B is empty. Water is poured from A into B until the water levels in the two tanks are equal. The rectangular tanks have the dimensions shown: A has base 40 cm × 30 cm and height 55 cm; B has base 70 cm × 30 cm and height 45 cm. Find the common water depth in centimetres.